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c Oksana Shatalov, Spring 2016 1 9.2 First-Order Linear Differential Equations A first order ODE is called linear if it is expressible in the form y 0 + p(t)y = g(t) (1) where p(t) and g(t) are given functions. The method to solve (1) for arbitrary p(t) and q(t) is called The Method of Integrating Factors The Method of Integrating Factors Step 1 Put ODE in the form (1). Step 2 Find the integrating factor R µ(t) = e p(t)dt Note: Any µ will suffice here, thus take the constant of integration C = 0. Step 3 Multiply both sides of (1) by µ and use the Product Rule for the left side to express the result as (µ(t)y(t))0 = µg(x) (2) Step 4 Integrate both sides of (2). Note: Be sure to include the constant of integration in this step! Step 5 Solve for the solution y(t). c Oksana Shatalov, Spring 2016 2 2 EXAMPLE 1. Solve the following DE: y 0 − 3xy = −xex . EXAMPLE 2. Consider y 0 − 2y = cos(3t). 1. Find the general solution. 2. Find the solution satisfying the initial condition y(0) = −2/13.